17th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar
19-22 May 2003, Monterey, California
AIAA 2003-2116
AIAA-2003-2116
DEVELOPMENT OF CONTROL ALGORITHM FOR THE AUTONOMOUS GLIDING
DELIVERY SYSTEM
Isaac I. Kaminer† and Oleg A. Yakimenko‡
Department of Aeronautics and Astronautics, Naval Postgraduate School, Monterey, CA
Abstract
The paper considers the development and simulation
testing of the control algorithms for an autonomous
high-glide aerial delivery system, which consists of
650sq.ft rectangular double-skin parafoil and 500lb
payload. The paper applies optimal control analysis to
the real-time trajectory generation. Resulting guidance
and control system includes tracking this reference trajectory using a nonlinear tracking controller. The paper
presents the description of the algorithm along with
simulation results and ends with guidelines for the further implementation of the developed software aboard
the real system.
foil from the initial position to the desired landing
point. Finally, it is the responsibility of the control system to track this trajectory using the information provided by the navigation subsystem and onboard actuators.
In the past decade, several GNC concepts for gliding parachute applications have been developed and
published.13-22 Most of them were tested in a simulation
environment, some in flight test.
Some of the published papers include analytical
development of feasible trajectories. In Ref.23, for example, minimum glide angle and descent rate, minimum turn radius and maximum gliding turn rate, reachable ground area, minimum altitude loss problems were
considered and solved analytically for a simplified
three-degree-of-freedom (3DoF) model with a parabolic
lift-drag polar. The problems of the maximum time to
stall the glider at the predefined point in a horizontal
plane and a maximum range for the appropriate simplified model with constraints were also considered. Optimization included no wind scenario.
Assuming the wind velocity to be constant Ref.24
synthesized zero-lag control of the gliding parachute
system with the constant rate of descent. This interesting work uses a compound performance index blending
time, kinetic energy, and controls expenditure. The optimal problem was solved by applying Pontrjagin's
maximum principle.25 The resulting two-point boundary-value problem was solved using KrylovChernous'ko method.26
However, majority of the published results on
GNC system development for parafoils sue heuristic
rules and classical control algorithms. It is appropriate
to mention some of them here emphasizing a scale of
the system, sensor suite specs and control approach.
Probably one of the first GNC algorithms for small
ram-air parachutes is presented in Ref.13. Computer
programs were written using 6DoF models of small
parafoils, such as Ram-air, Para-commander, and Standard Round. The aerodynamic models were simplified,
but they included the non-linearities and severe constraints common to all parachutes as compared to aircraft. The algorithms used GPS position and velocity
data provided at 1Hz to control both toggle lines. Homing and circling algorithms were simulated using direc-
.
I. Introduction
Maneuverable ram-air parafoils1-3 are widely used today. The list of users includes skydivers, smoke jumpers and special forces. Furthermore, their extended
range (compared to that of round canopy parachutes)
makes them very practical for payload delivery. Recent
introduction of the Global Positioning System (GPS)
made the development of fully autonomous ram-air
parafoils possible. Moreover, ram-air parafoils are
nowadays considered a vital element of the unmanned
air vehicles (UAVs) capability and of International
Space Station (ISS) crew rescue vehicle.4-12
Autonomous parafoil capability implies delivering
the system to a desired landing point from an arbitrary
release point using onboard computer, sensors and actuators. This requires development of a guidance, navigation and control (GNC) system. The navigation subsystem manages data acquisition, processes sensor data
and provides guidance and control subsystems with
information about parafoil states. Using this information along with other available system data (such as
local wind profiles), the guidance subsystem plans the
mission and generates feasible (physically realizable
and mission compatible) trajectory that takes the para-
This paper is declared a work of the U.S. Government and is
not subject to copyright protection in the United States.
.
†
Associate Professor; Senior Member AIAA.
‡
Research Associate Professor, Associate Fellow AIAA.
1
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Another GNC system incorporating appropriate
data acquisition system was developed to support highaltitude-high-opening insertion into hostile environments and was supposed to be used with a 500lb parafoil. The GPS guidance mode was designed to steer a
glider to the LZ using only GPS data or GPS plus up to
eight additional sensors. The GPS data was provided at
1Hz and inertial sensors at 200Hz. Control steering was
accomplished by actuating one of the two steering controls, which allowed four flight modes: straight flight,
steer left or right, hard turn left or right, and flare. No
test results however are available so far.
Another project was reported by the German Aerospace Center. It aimed to identify behavior of a small
(200lbs) and extensively instrumented parafoil-based
system and to investigate GNC-concepts for the
autonomous landing.29,30 Three and four degree of freedom models were developed and parameters of these
models were identified during this study.
The world's largest parafoil was designed for the
U.S. Army by Pioneer Aerospace Corp. This was a part
of a contract undertaken by Pioneer and the U.S. Army
Soldier Systems Command and was tested in 20012002 at the U.S. Yuma Proving Grounds, AZ (YPG) as
an aid for ISS "lifeboat". The resulting parafoil was
ultimately intended to evacuate a seven-person crew
from the station in case of an emergency (Fig.1).11-12
The 1,600lb (50 times the weight of a typical personal parachute) 7,500sq.ft parafoil wing has the wingspan wider than a football field and a canopy area of
more than one and a half times that of the wings of a
747 jumbo jet and is designed to carry weights of up to
42,000lbs, the heaviest cargo ever transported by a
parafoil. The system is able to move anything from
supplies to armored vehicles into strategic areas. Other
applications could include mentioned recovery of highvalue assets for the space program, and humanitarian
aid, such as the delivery of food, supplies and medications to war-torn areas like Bosnia, Sarajevo, Afghanistan, and Iraq.
The Pioneer Aerospace Corporation is an industry
leader in parachute design. They have provided recovery parachutes for NASA's space shuttle, the Air
Force's B-1A Bomber and parachutes for other NASA
and Department of Defense programs. Since testing on
the Guided Parafoil Aerial Delivery System (GPADS)
program began in January 1994, Pioneer holds records
for the largest parafoil, highest altitude (22,000ft), and
heaviest weight (15,500lbs) airdropped.
The GNC system for GPADS7,8 was developed by
the European Space Agency.22,29-33 The Parafoil Technology Demonstrator program was set up and conducted under ESA to demonstrate feasibility of guiding
a large-scale parafoil autonomously to a predefined
point and to perform a flared landing within a specific
range. Another objective was to enable further investi-
tion sines and cosines. Wind was included in all algorithms. Homing algorithms were included that increased the allowable offset, minimized the miss distance and reduced the impact velocity. These were
based on two-axis proportional navigation (PN), and
also upon upward-biased rate-integrating navigation in
glide angle. The latest produced final flare and nearly
vertical landing even from a long initial offset. For the
case of high initial altitude and small offset circling
algorithms were also introduced. Finally, combined
downwind, upwind, and guidance algorithms were
simulated to emulate manned parachute procedures.
Another approach was employed in Ref.16 where
Pseudo-Jumper (PJ) guidance and control algorithms
based on the professional jump procedures were introduced. The main phases of the mission after release
were defined as follows:
"Cruise" phase consisting of aiming for the holding
area way-points (WPs) by one or more types of homing algorithms;
"Hold" phase: As the parachute gets close enough to
the homing WPs, it is caused to perform S-turns just
downwind of the approach zone. The pattern is adjusted to suit the wind speed so that it stays near the
landing zone (LZ) and has just enough upwind distance left for the approach and flare;
"Center" phase: at some point, governed mainly by
altitude (time-to-go), the PJ must aim for the center
of the drop zone so that it is heading towards the
touchdown target. Enough time must be allowed to
complete any ongoing S-turn;
"Approach" phase: at some point, the PJ has to stop
trying to center and get lined up with the surface
wind direction and to minimize toggle actions past
that point;
"Flare" phase is triggered with rapid full braking to
arrest the vertical and/or horizontal inertial velocities
in an optimal control fashion.
Perfect knowledge of parafoil's relative location,
i.e. availability of a GPS sensor and of sonic altimeter
as well as of winds below 300ft was assumed.
Recent work by the Iowa State University
(http://cosmos.ssol.iastate.edu/RGS)27,28 and Atair
Aerospace
(http://www.extremefly.com/aerospace)
includes development of a parafoil-based Recovery
Guidance System to return payloads from a highaltitude balloon back to the ground safely and predictably. The parafoil is supposed to accompany the balloon
and to be cut free of the balloon at a predetermined
time. It then guides the payload to a designated LZ. As
reported, fully automated guidance hardware and software used DGPS (differential GPS) data, a digital compass, a linear quadratic regulator and PN guidance to
control and navigate the parafoil. Only simulation results have been reported so far.
2
gation into the flight dynamics of the parafoil systems.
The first automatically-guided, parafoil based descent
and landing of a test vehicle was carried out in support
of ESA's development in 1997. The spacecraft was
supposed to be completely automated, although the
crew will have the capability to switch to backup systems and steer the parafoil manually, if necessary.
Figure 1 The X-38 vehicle under the world’s largest parafoil (courtesy of NASA)
will lead to a touchdown at LZ. Seven guidance modes
are available:
Automatic trim;
Maneuver to WP;
Transition to hold;
Holding pattern (to maintain turn radius);
Maneuver to target;
Landing approach;
Landing flare.
The control system may switch between any of these
modes sequentially or bypass some of them.
The guidance concept presented in Ref.22 follows
generally the same scheme. The energy management
concept is adopted motivated by the way novice skydivers learn the landing approach.16 The trajectory is
planned using WPs. The guidance works as follows:
Homing: from the current position the energy management center (EMC) is approached if possible. If
the remaining distance is not long enough to reach
the desired impact point, the WP is moved in the direction to the final turn point (FTP);
Energy Management: after reaching EMC, surplus
altitude is reduced by flying S-patters to the energy
management turn points (EMTP). If the remaining
distance becomes too small to reach DIP, the EMTP
is shifted along the energy management axis towards
the EMC;
The prototype reference trajectory (RT) consisted
of the following nominal phases
Acquisition - trajectory initialization and initial turn;
Homing - travel to intercept Energy Management
Circle (EMC);
EMC Entry, EMC and EMC Exit - the energy management circling maneuver;
Final - final approach to target on desired heading;
The GNC system employs laser or radar altimeter
to trigger flare and is required to steer the parafoil to
within 100m of a preselected desired impact point (DIP)
(so far the accuracy of 200m was achieved on a smaller
scaled prototypes).
To support the development of GPADS software
and to evaluate its expected performance on ram-air
parafoils with payload capacities ranging from less than
200lbs to 40,000lbs, Draper Laboratory also developed
a GNC package.17-19 This software was demonstrated
using NASA Dryden Flight Research Center airdrop
test article known as the "Wedge 3" which utilizes an
88sq.ft ram-air parafoil and carries a payload of 175lbs.
Same GNC algorithms were also tested in simulation
using the model of a larger 3,600sq.ft guided parafoil
air delivery system with a 13,088lb payload. Upon full
deployment of the parafoil canopy the GNC algorithm
is activated. It determines the desired heading angle that
3
Landing: the landing phase is divided into 4 subphases: (1) transition into landing corridor, (2) approach of final turn point (FTP), (3) turn into wind
and (4) flare.
The Backup mode was also developed as follows:
if altitude and subsequently remaining distance are too
small to reach the landing point during homing phase,
the vehicle keeps heading the FTP until a specified decision altitude is reached. Then the vehicle turns against
wind independently of its position and prepares for
landing.
The guidance system output consists of a heading
command computed using current position and next
WP. For wind accommodation, the trajectory is planned
in the wind fixed coordinate system.21 Assuming constant horizontal wind, the actual position is shifted to a
virtual position by the predicted wind drift, the vehicle
will be affected during the remaining flight until touch
down. Within this wind fixed system the guidance can
be done as if there is no wind. Depending on the sign of
the heading error, either the left or the right actuator is
activated, leading to a left or a right turn. A saturation
limited PN controller computes the desired actuator
position. For instance, with an amplification of 4, a
commanded turn of 90° first activates one actuator to
full stroke until the heading difference goes below 15°
and the proportional part of the controller takes over.
According to Ref.22 after selective availability has
been switched off (in May 2000), plain GPS data without differential corrections is accurate enough for trajectory planning. The GPS altitude, with errors of about
±20m, is combined with the pressure altitude. Heading
information is supposed to be computed by applying a
complementary filter to blend azimuth and yaw rate as
described in Refs.14-15. Simulation results also show
that possible wind mispredictions have more influence
on the landing accuracy, than incorrect estimates of
model parameters and measurement errors. However, as
long as the errors stay in the normal range (position
±10m, altitude ±5m, heading ±10°) the GNC algorithm
is robust enough to land the vehicle within a range of
±50m around the predefined landing point, provided
that the wind is perfectly known.
As seen above in most GNC concepts, the guidance
itself is divided into three phases: homing, energy management and landing, where each of them can consist of
further subphases. Homing, i.e. flying towards the LZ,
and landing are similar in almost all guidance concepts.
Energy management is done by flying a certain pattern
close to the landing point, until some criteria is satisfied, switching to the landing phase. Most differences
among the realized guidance concepts are concerned
with the shape of the pattern and the criteria for the
landing decision.
Present paper addresses the problem of GNC development for a parafoil system as follows. First, a fea-
sible trajectory is generated in real-time that takes the
parafoil from initial release point to touch down at a
desired impact point. It is assumed that only the direction of the wind at LZ is known. The trajectory consists
of an initial glide, spiral descent and of final glide and
flare. The final glide and flare are directed into the wind
at LZ. This structure is motivated by optimal control
analysis carried out on a simplified model. The resulting trajectory is tracked using a nonlinear algorithm
with guaranteed local stability and performance properties (see Ref.34). The control algorithm converts trajectory-tracking errors directly into control actuator commands. Therefore, only GPS position and velocity are
needed to implement it.
As expected the basic trajectory structure is similar
to the ones reported in the literature. However, as
shown in Section III, a single smooth inertial trajectory
is generated using simple optimization and is tracked
throughout the drop by the same control algorithm. This
eliminates the need for multiple modes, extensive
switching logic and wind information throughout the
drop. The latter is made possible by the nonlinear control algorithm that tracks the inertial trajectory directly
and treats wind as a disturbance.
This specific delivery system considered in this paper is called Pegasus. It consists of 650sq.ft span rectangular double-skin parafoil and 500lb payload (Fig.2)
and was originally manufactured by the FXC Corp. It
can be controlled by symmetric and differential flap
deflections that occupy outer four (out of eight on each
side) cells of the parafoil.
This paper is organized as follows. Section II contains a brief description of the system and of the complete 6DoF model used to support the development of
the GNC algorithm. Section III introduces optimal control strategy for a simplified parafoil model using
Pontrjagin's principle of optimality. Section IV discusses real-time trajectory generation for the control
algorithm. Section V discusses the development of the
tracking control algorithm. Section VI introduces simulation results of the complete guidance and control system. Finally, Section VII contains the main conclusions.
II. System Architecture and Modeling
To support the development of the GNC algorithm, a
complete 6DoF model of Pegasus was developed and
tuned using flight test data provided by the YPG. Fig.3
outlines Pegasus dimensions.
The description of the 6DoF model used is given in
Ref.35. This model matches flight test data and is characterized by the following integral parameters: the average descent rate is 3.7-3.9m/s, glide ratio is about 3.0,
the turn rate of ~6°/s corresponds to the full deflection
of one flap. An interesting feature from the control
standpoint is that the system exhibits almost no flare
capability (flaps deflection results in almost no change
4
in the descent rate). Therefore, it is very important to
land it into the wind.
Figure 2 Double-skin 650sq.ft parafoil with 500lb payload
Figure 3 The 500lb parafoil geometry
5
III. Optimal Control Synthesis
Consider the following kinematic model of a parafoil in
the horizontal plane. Suppose we have a constant glide
ratio and by pulling risers we can control its yaw rate.
Mathematically, this is expressed by the following simplified equations:
x = V cosψ , y = V sinψ , ψ = v ,
(1)
This differential equation gives two sinusoids
(shifted with respect to abscise axis by ±Ξ −1 ) as solutions for the general (non-singular) case
pψ = C1 sin(Ξt + C2 ) ± Ξ −1 ,
(6)
where v ∈ [ −Ξ; Ξ ] is the only control.
moves along a descending spiral. It takes 2πΞ −1 seconds to make a full turn with a radius of V Ξ −1 . If
C1 = Ξ −1 there exists a possibility of singular control.
This is caused by the fact that there exists a point in
time where both pψ and pψ are zero as can be seen
where C1 and C2 are constants defined by the concrete
boundary conditions. If C1 ≠ Ξ −1 the parafoil model
The Hamiltonian for the system (1) for a timeoptimal control can now be written as:
 cosψ 
H = pψ v + V ( px , p y ) 
(2)
 −1 ,
 sinψ 
from (6).
Consider singular control for this model. By definition it means that pψ ≡ 0 . For the time-optimal prob-
where equations for adjoint variables px , p y and pψ
are given by
px = 0,
p y = 0,
lem from the Hamiltonian (2) and third equation in (3)
(of course keeping in mind the first two) it follows that
for a singular control case
px = V −1 cosψ , p y = V −1 sinψ , ψ = const . (7)
(3)
 sinψ 
pψ = V ( px , p y ) 
.
 − cosψ 
The optimal control for the time-minimum problem
now is given by
v = Ξsign ( pψ ) .
(4)
Expressions (7) imply that singular control corresponds to motion with a constant heading ( v ≡ 0 ). It
may not however be realized. Instead, the parafoil
model may switch from right-handed spiral to a lefthanded one or vice versa. Planar projections of possible
trajectories are shown on Fig.4. Concrete boundary
conditions define which case will be realized.
By differentiating the expression for pψ (3) and
combining it with Hamiltonian (2) for both cases when
pψ > 0 and pψ < 0 we can get a set of equations for
pψ :
pψ + Ξ 2 pψ ∓ Ξ = 0 .
(5)
Figure 4 Planar projections of possible trajectories
To summarize, Figs.5,6 show examples of threedimensional (3D) time-optimal trajectories computed in
the inertial frame for the gliding system (1). As suggested by the Pontrjagin's principle of optimality they
would consist of helices (spiral descent) and straight
descent segments. The basic difference between two
trajectories shown on Fig.5 is whether parafoil expends
its potential energy at the beginning of descent or at the
end. It may depend on the tactical conditions in the LZ,
terrain, or some other factors. In a general case parafoil
could spend its potential energy in the vicinity of some
other WP differing from the start and final portions of
the trajectory as it shown in Fig.6. In any case, since the
control system cannot meet any time constraints due to
6
unavailability of thrust on a parafoil system, the common feature of these RTs is that they are defined in the
inertial frame and are time independent.
Figure 5 3D representation of possible inertial RTs
Figure 6 General-case trajectory
Another critical issue is that in order to meet a softlanding requirement the parafoil at the LZ must align
itself into the wind.
ment. The radius of the spiral is adjusted to provide
smooth transition between each segment. The rest of
the section derives the mathematical representation of
the complete RT.
Let γ RT1 denote the desired flight path angle for the
IV. Real-Time Trajectory Generation
The optimal control analysis of the previous section
motivated the basic RT structure as is shown in Fig. 7.
This RT consists of three segments: initial straight-line
glide (segment 1), spiral descent (segment 2) and final
glide and flare (segment 3). Since it is required that the
parafoil is aligned into the wind at LZ, the final glide
segment ends at the desired impact point (DIP) and is
directed into the wind. Furthermore, the final glide
starts an offset distance d offset away and a certain height
first straight-line segment of the trajectory, similarly let
γ RT2 denote the flight path angle for the spiral-descent
segment and, finally, let γ RT3 represent the flight path
angle for the final straight-line segment. Furthermore,
let ψ f represent the wind direction at the LZ,
p 0 = ( x0 , y0 , z0 ) is the RP, and p f = ( x f , y f , z f
)
notes
vector
T
the
DIP.
Define
T
de-
i = ( cos(ψ f + π ), sin(ψ f + π ), 0 ) to be the unit vector
above DIP. The height is determined by the flight path
angle of the segment 3. Similarly, the first segment
starts at the parafoil release point (RP) and ends at a
point defined by the flight path angle of the first segment at the spiral descent segment. The first and last
segments are fused together by the spiral descent seg-
T
that points into the wind at the LZ and vector
i ∗ = ( − sin(ψ f + π ), cos(ψ f + π ), 0 ) to be the unit vecT
7
tor
orthogonal
to
i.
Then
the
initial
(
point
p3 = −d offset i + 0, 0, z f + d offset tan γ RT3
p3 = ( x3 , y3 , z3 ) of the third segment is given by
T
)
T
.
(8)
Figure 7 Graphical representation of the parafoil trajectory
Now, let p RT denote the position on the RT, s denote the path length traveled along the RT and VRT parafoil's velocity along this path. Then the expression
for the third segment of the desired trajectory is
p RT ( s ) = p 3 + T3 s , s = VRT ,
(9)
where
(
T3 = − cosψ f cos γ RT3 , − sinψ f cos γ RT3 , − sin γ RT3
)
T
Segment 3
pf
Segment 2
ψ2
(10)
ψ1
Then, along the segment 2 ( s1 < s ≤ s1 + s2 )
∗
 r sin ( ρ RT
( s − s1 ) + ∆ ) 


∗
p RT ( s ) = p1 +  −r cos ( ρ RT
( s − s1 ) + ∆ )  , (17)




( s1 − s ) sin γ RT2


∗
−1
∗
where
ρ RT = ψ RT VRT ,
∆ = ψ f + π − ρ RT ( s2 − s1 ) ,
T
p1 = p 3 + ri∗ ,
(12)
which is used next to determine the expression for p RT
s2 =
along segment 1. Let d = ( x0 − x1 ) + ( y0 − y1 ) , then
2
. (13)
Then
p 2 = p 0 + T1
d 2 − r2
, s1 = p 0 − p 2
cos γ RT1
z 2 − z3
.
sin γ RT2
Without loss of generality, the z-component of the
center of the spiral descent p1 was set to equal z2 in
the definition of p RT ( s ) above.
Let
∗
 cos ( ρ RT
( s − s1 ) + ∆ ) cos γ RT2 
dp RT


∗
T2 ( s ) = ds =  sin ( ρ RT
( s − s1 ) + ∆ ) cos γ RT2  .(18)


dp RT


− sin γ RT2
ds


Then, the choice of ∆ guarantees that the horizontal projections of vectors defined by Eqns. (18) and (10)
are aligned, i.e.
y − y0
r
, ψ 1 = tan −1 1
and ψ 2 = ψ 1 + ∆ψ .
d
x1 − x0
Analogous to (10) define
∆ψ = sin −1
T
p2
Figure 8 Top view of the parafoil RT
p1 = ( x1 , y1 , z1 ) has the following expression
)
∆ψ
r
Segment 1
first segment and s2 - of the second. Then, along the
third segment ( s1 + s2 < s ≤ s1 + s2 + s3 )
p RT ( s ) = p 3 + T3 ( s − s1 + s2 ) .
(11)
Next, consider Fig.8. Let r represent the radius of
the spiral descent segment. Then the center of the spiral
(
d
p0
s3 = p 3 − p f . Similarly, let s1 denote the length of the
T1 = cosψ 2 cos γ RT1 ,sinψ 2 cos γ RT1 , − sin γ RT1
p1
North
is the unit vector that points along the line connecting
p3 to p f . Note, that the length of the third segment
2
p3
(14)
and
p RT ( s ) = p 0 + T1 s , 0 ≤ s ≤ s1 .
(15)
The turn rate ψ RT along the spiral descent segment
is given by
ψ RT = VRT r −1 cos γ RT2 .
(16)
T2hor
8
s = s2
= T3hor
s = s2
= ( − cosψ f , − sinψ f
)
T
. (19)
shown in Ref.34 trimming trajectories consist of
straight lines and helices. Therefore, this methodology
is particularly suitable for the problem at hand.
The key ideas of the design methodology developed in Ref.34 include the following five steps:
1) reparameterize trimming trajectory using the arclength s, thus eliminating time as an independent
variable;
2) resolve the position and velocity errors in so-called
Frenet frame (Ref.34);
3) form error dynamics for the system consisting of
the trajectory and parafoil model, where the position and velocity error states are resolved in Frenet
frame;
4) design a linear tracking controller for the linearization of the system along the trimming trajectory;
5) implement this controller with the true nonlinear
plant using a nonlinear transformation provided in
Ref.34. This implementation guarantees that the
linearization along the trimming trajectory of the
feedback system consisting of the nonlinear plant
and nonlinear controller preserves the eigenvalues
and transfer functions of the feedback interconnection of the linearized error dynamics and linear
tracking controller (see Ref.36).
This fact implies that at p3 the horizontal projections of the commanded velocity vectors for segments 2
and 3 are equal and are independent of the choice of r.
Finally, the radius of the spiral descent r is selected
to guarantee that at p 2 segments 1 and 2 merge
smoothly. This is done numerically by solving a single
variable constrained optimization problem. Note that at
p2
p RT
= p 2 = p1 + ( r sin ∆, − r cos ∆, 0 ) .
T
s = s1
(20)
Therefore, let
e( r )
p 2 − p1 + ( r sin ∆, − r cos ∆, 0 )
T
,
(21)
then the desired value of r is rdes = arg min e(r ) . Fig.9
rmin ≤ r ≤ rmax
includes a plot of e(r) versus r, which indicates that
there are multiple values of rdes . Therefore the values of
rmin and rmax are selected to provide a unique solution
that is consistent with physical limitations of the parafoil. Note, in steady state turn the bank angle of the
parafoil is defined by the following expression:
 VRT2 cos γ RT2 
−1 VRTψ RT
−1
= tan 
φRT = tan
 , (22)


g
rg


where g is acceleration due to gravity. By construction
the horizontal projections of T1 and T2 at p 2 are
aligned, i.e. T1hor
s = s1
= T2hor
s = s1
V.1 Derivation of the errors in Frenet frame
As shown in Section IV the desired RT is parameterized using the path/arclength parameter s.
.
Let p = ( x, y, z )
T
denote current position of the
parafoil. The methodology developed in Ref.34 requires
that position and velocity commands p RT ( s∗ ) , VRT ( s∗ )
used to compute position and velocity errors,
p RT ( s∗ ) − p and VRT ( s∗ ) − V respectively, correspond
to the point on the RT that is nearest to current position
of the vehicle. This is done by first determining the
value
of
the
path/arclength
parameter
s∗ = arg min p RT ( s ) − p
2
and then using it to compute
s
position and velocity commands. In Ref.34 the problem
of determining s∗ is reduced to a constrained optimization problem. However, computing limitations of the
onboard processor have imposed a need to develop analytical techniques for the computation of s∗ , discussed
next.
An exact analytical expression for s∗ can be derived for straight-line segments 1 and 3. Recall Eqns.
(15) and (11). Then
Figure 9 A representative plot of e(r) versus r
V. Integrated Guidance and Control Algorithm
The development of the integrated guidance and control
algorithm is based on the work reported in Ref.34,
where authors have proposed a new technique for tracking so-called trimming trajectories for UAV's. As
9
 p 0 + T1 s − p ,

0 ≤ s ≤ s1

(23)
s∗ = arg min 
 p 3 + T3 ( s − s1 − s2 ) − p , s1 + s2 < s ≤ s1 + s2 + s3 
s


Simple algebra shows that for straight-line seg
 x1 − xcyl

−1
ments
+π 
 xΠ   r tan 
  . (28)
Π (p cyl ) =:   = 
 y1 − ycyl

T1T (p − p0 ), 0 ≤ s ≤ s1 ,
z
 Π  

s∗ =  T
(24)
z
cyl


T3 (p − p3 ) + ( s1 + s2 ), s1 + s2 < s ≤ s1 + s2 + s3 .
Π
maps
the
cylinder
C
onto
a recThe
function
Now, recall that segment 2 represents a spiral. In
tangle
of
width
2
π
r
and
height
z
−
z
.
Moreover,
the
2
1
this case, only an approximation of s∗ can be found
trajectory p RT ( s ) defined for segment 2 is mapped into
analytically. Its derivation is discussed next.
Let
a function Π (p RT ( s )) shown in Fig.10.
C = {(l , m, n) : (l − x1 ) 2 + (m − y1 ) 2 = r 2 , z1 ≤ n ≤ z2 } (25)
Let d∗ denote the distance from p to the nearest
T
define a cylinder centered at p1 = ( x1 , y1 , z1 ) . Then the
point p RT ( s∗ ) on the trajectory p RT ( s ) . Then p RT ( s∗ )
can be obtained by determining the intersection of the
spiral descent trajectory
∗
ball Bd∗ of radius d∗ centered at p. The mapping
 r sin ( ρ RT ( s − s1 ) + ∆ ) 


Π (Bd∗ ∩ C ) is an ellipse centered at Π (p proj ) with
∗
p RT ( s ) = p1 +  −r cos ( ρ RT
( s − s1 ) + ∆ )  (26)


semi-major and semi-minor radii given by rz = d x 3


−( s − s1 ) sin γ RT2


r 2 + (r + d x ) 2 − 4d x2
and rx = r cos −1
, respectively. In
( s1 < s ≤ s1 + s2 ) is a subset of C. Furthermore, the pro2r ( r + d x )
jection of the vehicle's position p onto C is
the previous expressions d x = ( x − x proj ) 2 + ( y − y proj ) 2 .
T

( x − x1 , y − y1 ) 
,z
p proj =  ( x1 , y1 ) + r
(27)
Intersection of this ellipsoid with Π (p(s )) represents


x − x1 , y − y1


the mapping Π (p( s∗ )) of p( s∗ ) . Analytical approximaT
For any vector p cyl = ( xcyl , ycyl , zcyl ) ∈ C define a
tion of Π (p(s∗ )) is obtained next and then used to com3
2
pute s∗ .
function Π : R → R ,
Π
zΠ
z2
p RT ( s )
p proj
Π (p RT ( s ))
p
Π ( Bd∗ ∩ C )
Π (p proj )
γ RT
2
z1
2π r
Cylinder C
xΠ
Figure 10 Graphical representation of Π (C )
Consider Fig.11. Note for the case illustrated in
this figure, the zΠ coordinate of Π (p( s∗ )) is bounded
above by z1n and below by z1 p . Using the basic ge-
z1n = z proj + mod(z0 n − z proj , 2π r tan γ RT2 ) ,
z1 p = z proj + z1n − z proj cos 2γ RT2 ,
where
ometry shown in Fig.11, expressions for z1n and z1 p
are:
10
(29)
z0n = z1 − s∗n sinγ RT2 ,
x

∗  proj
s∗n = ρ RT
− mod(∆, 2π )  .

r


Similarly,
z2 n = z1n − 2π r tan γ RT2 ,
(30)
z2 p = z proj − z2 n − z proj cos 2γ RT2 .
(31)
zΠ
z2
z0n
Π (p RT ( s ))
γ RT
Π (p( s∗ ))
2
z1 p
Π (p proj )
z1n
z1
z2 p
z2n
2π r
xΠ
Figure 11 Graphical representation of the range of the function Π
The variables z2 p
and z1 p
∗
 cos ( ρ RT
( s∗ − s1 ) + ∆ ) cos γ RT2 


∗
T2 ( s∗ ) ==  sin ( ρ RT
( s∗ − s1 ) + ∆ ) cos γ RT2  ,




−
sin
γ
RT2


represent the z-
coordinates of the projections of Π (p proj ) onto the two
nearest legs of Π (p( s )) . These variables can now be
used to find the approximation of zΠ , zapp :
(
∗
 − sin ρ RT
( s∗ − slim1 ) + ∆

∗
N 2 ( s∗ ) ==  cos ( ρ RT
( s∗ − s1 ) + ∆ )


0

T ( s ) × N 2 ( s∗ )
B 2 ( s∗ ) = 2 ∗
.
T2 ( s∗ ) × N 2 ( s∗ )
 z1 p ,
if z1 p − z proj < z2 p − z proj ,

(32)
zapp = 
 zapp = z2 p , otherwise.
Note that logic (32) guarantees that in the case
when two legs of p( s ) are equidistant from p, the point
closer to the ground is selected. Furthermore, since Π
maps z-axis of R 3 onto itself zapp can be used to comz1 − zapp
sin γ RT2
,
,



(35)
For segments 1 and 3 the basis vectors are constant. For example, for segment 1 we obtain
 cosψ 2 cos γ RT1 
 − sinψ 2 




T1 =  sin ψ 2 cos γ RT1  , N1 =  cosψ 2  . (36)


 0 


 − sin γ RT1 
pute s∗
s∗ =
) 
(33)
which follows from the definition of p( s ) for the spiral
descent segment. This value of s∗ can now be used to
compute the unit basis vectors T2, N2, and B2 of the
Frenet frame {F} for segment 2 at s = s∗
dp c
dT2
T × N2
T2 = ds , N 2 = ds , B 2 = 2
(34)
dp c
dT2
T2 × N 2
ds
ds
or in the final form
And for segment 3,
 − cosψ f cos γ RT3 
 sinψ f 




T3 =  − sinψ f cos γ RT3  , N 3 =  − cosψ f  . (37)




0
− sin γ RT3




The basis vectors B1 and B3 are computed exactly
as B2. Now the rotation matrix LTPF Ri from LTP to Frenet frame for the i-th segment is given by
F
LTP
Ri = ( Ti , N i , Bi )
T
and the corresponding position
and velocity errors are
11
pe =
F
LTP
Ri (p RT ( s∗ ) − p) =: ( 0, ye , ze ) ,
pe =
F
LTP
Ri (p RT ( s∗ ) − V ) +
only use GPS position and velocity for feedback. This
constraint is motivated by the requirement to keep the
cost of the onboard avionics low, therefore, only GPS is
available for control.
During segments 1 and 2 the control system uses
differential flaps to drive the lateral components of the
position and velocity error vectors to zero. On the other
hand, the last segment is tracked using both symmetric
and differential flaps, denoted here by δ S and δ D , respectively. The structure of tracking controller for the
lateral channel shown is Fig.12.
T
F
LTP
Ri (p RT ( s∗ ) − p).
(38)
Notice that by construction the x-component of the position error vector p e is zero, see Ref.34. These error
vectors were used to design a tracking control system
discussed in the next section.
V.2 Control System Design
The trajectory tracking controller design methodology
proposed in Ref.34 is now applied to the design of a
tracking controller for the trajectory generated using the
algorithm developed in Section IV. The controller can
p
Computation
of the position
and velocity error
vectors
Reference
trajectory
generator
Ky
ye
δD
Ky
trajectory
parameters
-
V
ye
T −1
z −1
Figure 12 Lateral channel tracking controller structure
used to track the final segment of the trajectory. This
decision was motivated by the fact that symmetric flaps
have negligible control authority in spiral descent. Furthermore, deflecting symmetric flaps tends to increase
the descent rate of the parafoil while reducing the flap
deflection budget available for differential flaps command and, therefore, symmetric flaps were not used
during segment 1 as well.
The gains K y and K y shown in Fig.12 were selected to provide stability and performance for the lateral channel of the feedback system. The variables z −1
and T denote backward shift and sampling period. The
sampling period for this problem was dictated by the
onboard GPS receiver, whose update rate is 2Hz.
The structure of the vertical channel controller is
shown in Fig.13. This controller uses symmetric flaps
δ S to drive the vertical channel error to 0 and was only
p
V
Reference
trajectory
generator
Computation
of the position
and velocity error
vectors
ze
ze
PID controller
δS
trajectory
parameters
Figure 13 Vertical channel tracking controller structure
son between flight test data collected during Pegasus
drop at the YPG and model's response to the wind collected during the same drop. Clearly, the 6DOF model
provides a reasonable approximation of the parafoil
dynamics.
VI. Simulation Results
The trajectory tracking control system discussed in Section V has been tested in simulation using a nonlinear
6DOF model of the Pegasus parafoil that also includes
the available actuator model.35 Fig.14 shows a compari-
12
Fig.15 shows results of an extensive simulation
analysis of the trajectory tracking control system using
the 6DOF model discussed above. Each of 40 simulation runs included the wind profile obtained at YPG.
The GPS position and velocity errors were modeled as
white noise processes with zero mean and standard deviations shown in Tab.1. Finally, landing performance
statistics shown in Tab.2 indicate that system's performance exceeded the required circular error probable
(CEP) of 100m.
Table 1 GPS errors (STD)
x-direction y-direction
Position (m)
5
5
Velocity (m/s)
0.3
0.3
CEP (m)
70.7
Figure 14 Comparison of the flight test data and simulation results
z-direction
10
0.5
Table 2 Touchdown errors
Mean value (m)
STD (m)
67.5
20.5
Altitude (m)
3000
2000
1000
-2000
-1500
0
2000
-1000
DIP
1000
-500
0
0
-1000
-2000
East-West (m)
500
South-North (m)
Figure 15 Simulation results of the trajectory tracking control system
2.
VII. Conclusions
The optimal control strategy for the Pegasus parafoil
payload delivery system was synthesized based on
Pontrjagin's maximum principle. This motivated the
structure of the real-time reference trajectory generator.
Together with the robust path following algorithm it
enabled a successful development and simulation testing of the complete guidance and control algorithm.
The authors plan to flight test developed algorithms in
the early spring of 2003 and to have complete flight test
results soon afterwards.
3.
4.
5.
6.
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